MNT: Deprecate rcParams._get("backend")

since 3.10 we have the official API `matplotlib.get_backend
(auto_select=False)`. No problems have been reported and
matplotlib-inline is using it https://github
.com/ipython/matplotlib-inline/pull/38. Therefore, I remove the
provisional status from the auto-select flag.

Additionally, I deprecate the API rcParams._get('backend'), which the
above is replacing.

It seems that `rcParams._get("backend")` is not used anywhere
publically, but let's still be defensive and deprecate.

https://github.com/search?q=%2F%5C._get%5C%28%5B%22%27%5Dbackend%5B%22%27%5D%5C%29%2F+language%3APython+NOT+is%3Afork+NOT+path%3A**%2Fmatplotlib%2F**+NOT+path%3A**%2Fsite-packages**+NOT+path%3A**%2Fpyplot.py&type=code

This is working towards matplotlib#26406.
Follow-up to matplotlib#29039.

Author: timhoffm

doc/api/next_api_changes/deprecations/30963-TH.rst

@@ -0,0 +1,7 @@
+``rcParams._get("backend")``
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+``rcParams._get("backend")`` is deprecated. Instead, use the public API
+``matplotlib.get_backend(auto_select=False)`` to retrieve the current
+backend without triggering backend resolution. This API is available since
+Matplotlib 3.10.

lib/matplotlib/__init__.py

@@ -712,6 +712,12 @@ def _get(self, key):
 
         :meta public:
         """
+        if key == "backend":
+            _api.warn_deprecated(
+                "rcParams._get('backend') is deprecated since Matplotlib 3.11. "
+                "Use matplotlib.get_backend(auto_select=False) instead, which is "
+                "available since Matplotlib 3.10.",
+            )
         return dict.__getitem__(self, key)
 
     def _update_raw(self, other_params):
@@ -1278,23 +1284,14 @@ def get_backend(*, auto_select=True):
 
         .. versionadded:: 3.10
 
-        .. admonition:: Provisional
-
-           The *auto_select* flag is provisional. It may be changed or removed
-           without prior warning.
-
     See Also
     --------
     matplotlib.use
     """
     if auto_select:
         return rcParams['backend']
     else:
-        backend = rcParams._get('backend')
-        if backend is rcsetup._auto_backend_sentinel:
-            return None
-        else:
-            return backend
+        return rcParams._get_backend_or_none()
 
 
 def interactive(b):